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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, logarithmic growth describes a phenomenon whose size or cost can be described as a
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
function of some input. e.g. ''y'' = ''C'' log (''x''). Note that any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant.. Logarithmic growth is the inverse of
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
and is very slow. A familiar example of logarithmic growth is a number, ''N'', in
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which th ...
, which grows as log''b'' (''N''), where ''b'' is the base of the number system used, e.g. 10 for decimal arithmetic. In more advanced mathematics, the partial sums of the harmonic series :1+\frac+\frac+\frac+\frac+\cdots grow logarithmically. In the design of computer
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s, logarithmic growth, and related variants, such as log-linear, or
linearithmic In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
, growth are very desirable indications of efficiency, and occur in the
time complexity In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
analysis of algorithms such as binary search. Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll. It also plays a role in the St. Petersburg paradox. In
microbiology Microbiology () is the scientific study of microorganisms, those being unicellular (single cell), multicellular (cell colony), or acellular (lacking cells). Microbiology encompasses numerous sub-disciplines including virology, bacteriology, ...
, the rapidly growing exponential growth phase of a
cell culture Cell culture or tissue culture is the process by which cells are grown under controlled conditions, generally outside of their natural environment. The term "tissue culture" was coined by American pathologist Montrose Thomas Burrows. This tec ...
is sometimes called logarithmic growth. During this
bacterial growth 250px, Growth is shown as ''L'' = log(numbers) where numbers is the number of colony forming units per ml, versus ''T'' (time.) Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing ...
phase, the number of new cells appearing is proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis..


See also

* (an even slower growth model)


References

{{DEFAULTSORT:Logarithmic Growth Logarithms